Semicond. Sci. Technol. 9 (1994) 1775-1780. Printed in the
1I
UK
Excitons in parabolic quantum dots in
electric and magnetic
fields
S Jaziritl: and R Bennaceurl
7 Departement dePhysique d e I’Ecole Normale Superieure de Bizerte, Jarzouna,
7021 Bizerte, Tunisia
f Laboratoirede Physique de la Matiere Condensee, Faculte des Sciences deTunis,
1060 Le Belvedere Tunis, Tunisia
Received 16 February 1994, in final form 8 June 1994, accepted for publication
13June1994
Abstract. We report a perturbative-variational calculation on the effect of
magnetic and electric fields on excitons in parabolic quantum dots. The effects of
electric and magnetic fields on 1 s exciton energy and oscillator strength are
investigated. The competition between the confinementand correlation effects on
the one hand, and the external field effects (electric and magnetic) on the other
hand, is also discussed
1. Introduction
The development of semiconductor nanostructures, such
as quantum boxes and dots, by the use of different
techniques, has attracted much theoretical and experimental attention [l-123. In undoped quantum dots the
optical properties are dominated by excitonic effects. In
this paper, we consider quantum dots exhibiting a nearly
parabolic confinement for both electrons and holes [IO,
It]. These quantum dots can be fabricated by focused
laser-beam-induced interdiffusion of a GaAs/GaAlAs
quantum well system. For such a confinement, for both
electrons and holes, the centre of mass and relative
motion of the exciton separate [7, 121. In this case, the
motion of the exciton in the ground state is essentially
governed by the relative Hamiltonian [7, 121.
The application of external fields (magnetic [7j or
electric [ S . 121) influences exciton properties. The
difficulties encountered in theoretical calculations arise
from the need to include four different potentials
governing exciton motion: these are the confinement,
Coulomb, magnetic and electric potentials. Excitons in
quantum boxes have been studied by Bryant [2], and
have revealed the competing effects of quantum confinement and electron-hole correlation induced by the
Coulomb potential. The influence of a magnetic field on
excitons in parabolic quantum dots has been analysed
by Halonen et al [7]. This work shows that the erect
of the magnetic field is decreased when the confinement
potential is increased and vice versa. The quantum
confined stark erect (QCSE), induced in excitons by an
electrostatic field, has been studied in spherical [SI and
in parabolic [ l 2 j quantum dots. In particular, it is shown
in [12] that quantum dots with a low confinement
potential are more affected by the electric potential and
exhibit a larger Stark shift. On the contrary. quantum
0268-1242/94/101775+06$19.50@ 1994 IOP Publishing
Lid
dots with strong confinement have the smallest Stark
shift. The increase of confinement potential reduces the
effect of the electric field. Therefore, it is interesting to
study these effects on excitons in GaAs quantum dots,
which exhibit an important quantum-size potential (with
the radius of quantum dots ranging from SO to 200 A).
In the present paper we shall present theoretical results
on the properties of excitons in a parabolic quantum dot
in the presence of electric ( F ) and magnetic ( B ) fields,
applied parallel to the Iaxis. In our calculation we use
an analytical approximation by means of a perturbativevariational approach [9, 131, where the Coulomb
interaction potential is approximated by a force interacting according to Hooke’s law [6, 121.
2. nne!hod of ca!cu!l!!nn
Our calculation is based on the effective-mass approximation and ignores vaIence band structure effects. For
simplicity we consider a GaAs quantum dot with
parabolic confinement for the electrons and holes having
the same quantization energy hi2 [6, 10-123. The total
Hamiltonian for the exciton in a parabolic quantum dot
in the presence of electric ( F ) and a magnetic ( B ) fields
applied in the same direction (z axis), can be expressed as
H
= H,
+ Hh --e’E r
and
1775
S Jaziri and R Bennaceur
where H, (Hh) and m, (mh) are the single-particle
Hamiltonian and the effective mass for the electron (hole)
and E is the GaAs background dielectric constant. In
order to solve this total Hamiltonian we use the relative
coordinate I' = ( I ' ~- rh) and the corresponding momenta
p with reduced mass p (=m,m,/(m, + mh)) and centre-ofmass coordinate R (=(merh + mhrh)/(m, + mh)) and the
corresponding momenta P with the total mass M =
m,
inh. For the magnetic vector potential, we consider
the symmetric gauge vector potentials for the electron
and hole as [7] A , = @ x (re - r h ) and A h =
- $ E x (r, - vh) where the magnetic field is parallel to
the x direction. Therefore the total Hamiltonian is
+
H
+ 4.+ Hcoup
= HR
P2
HR = 2M
+Z I M Q ~ R ~
- _e2-
eFi
to a frequency a,. The io shift reflects tendency to a
breaking of the electron-hole pair by the field, while the
frequency Q, accounts for the remaining Coulombic
attraction between the spatially separated electron and
hole. In the (x,y) plane the electron-hole pair energy
will also be lowered by the Coulombic term. Without
Coulombic attraction the electric field would shift the
electron and hole mean positions by zo (zo = -eF/pQ2)
without altering the oscillation frequency (but giving rise
to a quadratic negative shift -e2F2/2pQ2). The application of the magnetic field HL can be described by
diamagnetic and Zeeman terms. The latter term makes
no contribution to the exciton ground state. In the (x,y)
plane, the diamagnetic term added to the parabolic
confinement potential enhances the exciton energy.
To apply the previous physical reasoning analytically
we add to and subtract from the relative electron-hole
Hamiltonian H, a term V(r) in order to split H, into two
parts. The first one Ho should be analytically solvable
while the second one H,, expected to be small, will be
treated by first-order perturbation calculus on the
eigenstates of H,,. There we choose [6, 121
&I
PI,,,,
=
V(r) = j&u(nzr2
ihe
BI' x V R
- hQ)
(34
where E. is still an unknown parameter. In this case
Me
where pR = he/2pc is the Bohr magneton,
gP
=
(i 2)
-
is the g factor and o = Be/pc is the cyclotron
frequency. The operator ipBg,,B(a/atp) corresponds to the
z component of the angular momentum operator.
The last term in the Hamiltonian, HEovp,
couples the
centre-of-mass and relative motions. According to [7,
141, this crossed term leads to a negligible perturbation
of the ground exciton energy, and it couples only excited
states of different relative and centre-of-mass motions
and is neglected in the following. We can see that the
exciton properties in the ground state can be essentially
determined by the relative Hamiltonian H,. The relative
Hamiltonian describing the exciton motion is
H, = Hl
+ H5 + H,
The perturbed term H , restores the character of the
Coulomb interaction. The zeroth-order wavefunction
and eigenenergy of the ground state of Ho are determined
exactly. This is possible because the unperturbed eigenfunctions are separable and analytically expressed as
x(R,r, E. F,2) = Q(R)$(p,j.)O(z,lJ
=Q
(R)(i>(L)'"
RI&
R,&
x exp(-pz/2R$) exp( -zZ/2Rf)
(4)
where
where pL and p z are respectively the transverse and
longitudinal electron-hole relative momenta. In the
absence of the magnetic field, the combined effects of the
Coulombic potential, the electric field term -eFz and
the parabolic confinement along z qualitatively result in
the build up of a nearly parabolic potential along z
displaced by io from the original one and corresponding
1776
RO
(1 + J.),'4'
The first term in (4) is the exciton centre-of-mass
wavefunction for a harmonic oscillator and the last terms
correspond to the exciton relative motion: $ ( p , % ) is
Excitons in parabolic quantum dots
the two-dimensional exciton wavefunction and e(z, 1)the
one-dimensional exciton wavefunction.
We determine the best choice of 1. as the one which
ensures the fastest convergence of the perturbation series.
so that only the first few terms will be used. Under these
circumstances we shall write the total energy as
- i.hQ
+ AE(1)
(6)
where E , is the gap energy and AH.?.) is approximated
by the first-order energy matrix correction calculated by
the perturbative method. The unknown parameter A in
equations (3)-(6) is determined by minimization of the
total energy value by requiring that the first-order energy
shift AE(E.) = ( Y j IHllYA) vanishes, where Y;. is the
ground state of Ifo.
We calculate the oscillator strength of the exciton in
the ground state in the dot normalized to that of a free
exciton in a bulk material with volume V = $ltRi. The
normalized oscillator strength is given by [9]
where E,, and ET(F,B ) are the total exciton energies in
the bulk and quantum dot respectively and a3Dis the
bulk effective Bohr radius.
3. Results and discussions
Results are presented for excitons confined in GaAs
parabolic quantum dots placed in electric and magnetic
fields parallel to the z direction, with the parameter
values: gap energy E, = 1.520 eV. dielectric constant
e = 13.1. effective masses for the electron (e), heavy hole
(hh),m, = 0.067m0,m,, = 0.377m0 (m, is the free-electron
mass).
Following the perturbative-variational method developed in [13], the validity of the calculation can be
tested by the dependence of the energy on the value of
1.We find that the energy will be independent of 71 if
the true wavefunction is used to calculate the groundstate energy. We also compare our results with those
obtained from conventional variational calculation for
the Hamiltonian defined in (1) by using a Gaussian trial
wavefunction of the type
Yver(r)
= N exp( -apz/2) exp(-flz2/2)
where N is the normalization factor and E , fl are the
variational parameters. Our results and the variational
results are very close except at large values of R , where
exciton energies calculated from our model are slightly
higher. However, our results and the conventional ones
are in agreement for R, < 200 A. For R, = 150 A and
F = 80 kV cm-I and B = 5 T the two methods give E,
values which differ by less than 0.2 meV. Our approximation is in fact coherent with the physics underlying
equations (3)-(5): the exciton motion is mostly dominated
by the confinement potential, the Coulombic term
essentially correlates the electron-hole reduced motion
in wider quantum dots. Also our approach is justified by
the relativemagnitude ofthe variational parameters E-'''
and fl-''z compared respectively with Rs(A) and R&)
values for R, = 100 A and for F = 80 kV cm-' and
B = 5 T; the deviation of our calculated values from the
variational results is 4%. It should be noted that the
perturbative-variational method is much simpler than the
variational method.
The effects of confinement, Coulomb, magnetic and
electric potentials can be written in terms of characteristic
length as
respectively. These effects on the properties of the excitons
can be classified with regard to the two intrinsic lengths:
the dimension of the parabolic quantum dot R, and the
excitonic Bohr radius a,D (the 3D Bohr radius = 140 8,
for GaAs). In this work we consider only quantum dots
with radii between 50 A and 200 8,where the confinement
effect is predominant. In this case the correlation effect
on excitons introduced by the Coulombic potential is
weak. In order to analyse the effects of magnetic and
electric fields on the energy E, and the oscillator strength
of the ground exciton state we define first a total (effective)
magnetic field E,,, =
where Bo =
is
an effective magnetic field associated with the 3D
parabolic quantum dot confinement, and LOff=
is the corresponding magnetic length. In the absence of
an applied electric field the increase in E, results in a
diamagnetic compression of the exciton wavefunction in
the (x, y ) directions. Under an applied electric field, the
distance between electrons and holes is increased and
induces a decrease of the energy E,. This decline of E,
results in the spatial separation of the exciton wavefunction along the iaxis.
In figure 1 we show the variation of the exciton energy
in the ground state as a function of the applied electric
field F for different quantum dot radii and values of the
magnetic field B: R , = 50 8, for B = 0 and B = 20 T;
R, = 120 8,for B = 0 and B = 10T and B = 20T. Note
that the application of the magnetic field reduces the
lateral dimension of the quantum dot corresponding to
L,, = 46 8,(for R , = 50 ,&and B = 20 T) and Le,, = 74 8,
(for R, = 120 8, and B = 10 T). We can see that for
narrow dots (R, < 1008,) the curvature of E T ( F ) is
negative, due to the quadratic Stark shift. For large
quantum dots ( R , > 100 8,) the curvature is negative at
low F and decreases more rapidly at large F. The reason
i's that for fields such that the interacting electron-hole
pair is well separated in the z direction, i.e. lzol > R,
(-8 x 10-8F (kV cm-')
> l), for this range
of fields and dot sizes the energy is approximately equal
to -e2/&lzoi.This regime cannot be reached in the case of
m,
1777
S Jaziri and R Bennaceur
1640
-
__Ros50A,B=O
cl:Ro=50A,B=20T
620
1600-1580-
%
E
___ Ro=l
'1560-
OAB 0
bRO=l2$.~=~OT
3
narrow dots, where the effect of the field F (F = 100 kV
cm- ') is not large enough; thus only the F*-like variation
appears in the E,(F) curves. The effects of the Stark shift
of the exciton in the quantum dot leads to a decrease of
the oscillator strength in figure 2. The decrease of the
oscillator strength in wider quantum dots can he
explained by the field-induced polarization of the
electron-hole wavefunction.
Figure 3 shows the effect of the magnetic field on the
exciton energy in quantum dots of different sizes for
different values of applied electric field. The increase in
the energy with increasing magnetic field is related to the
compression of the exciton wavefunction in the (x,y )
directions. In the absence of an electric field ( F = 0), the
application of the magnetic field creates an additional
confinemenf lowering the value of LFrr(the corresponding
magnetic length). We have two limiting situations: a
diamagnetic-like shift ET Bo(l B2/2B3 whenever
Bo % B and a high-field limit E,(B) % B(l + B$2B2)
when Bo @ B. For strongly confined systems (B 6 Bo) we
have very smooth parabolic dependencewith the external
field. For wider dots both limiting situations are present,
although the variations with B are sensibly smoothed by
the effect of geometrical confinement. Under the application of an electric field the electrons and holes begin
to separate. As the electrons and holes are pushed apart
by the electric field a reduction in exciton energy with
increasing electric field is expected. Figure 4 shows the
magnetic-field dependences of the normalized oscillator
strength. The increase in the oscillator strength with
+
1420
50
0
F (kV/cm)
Figure 1. Dependence of the energy of a 1 s exciton in a
GaAs parabolic quantum dot on the electric field for
different values of magnetic field and dot radius.
'
1640
--b :Ra=50A.B=20T
1620
4
1
Ro=S.OA.F=O
a:Ro=5OA.F=20kV/cm
---
'
---
Ro=lZOkB=O
c :Ro=l2OA.B=1 OT
d :Ro=l2OA.B=2OT
154.0
Figure 2. Dependence of the normalized oscillator
strength of a 1 s exciton in a GaAs parabolic quantum dot
on the electric field for differentvalues of magnetic field
and dot radius.
1778
Figure 3. Dependence of the energy of a 1 s exciton in a
GaAs parabolic quantum dot on the magnetic field for
different values of electric fieid and dot radius.
Excitons i n parabolic quantum dots
a
1
I
Ro=SOA.F=O
a:Ro=50A,F=ZOkV/cm
.
---
--LRo=120A,F=0
b R o = l Z O h . F = l Okv/cm
c:Ro=lZO&F=ZOkV/cm
Figure 4. Dependence of the normalized oscillator
strength of a 1 s exciton in a GaAs parabolic quantum dot
on the magnetic field for different values of electric fieid
and dot radius.
I
1660 I
50
150
100
Ro
(1)
200
Figure 5. Dependence of the energy of a 1 s exciton in a
GaAs parabolic quantum dot on the radius of the dot for
different values of electric and magnetic fields.
Figure 6. Dependence of the normalized oscillator
strength of a 1 s exciton in a GaAs parabolic quantum dot
on the radius of the dot for different values of electric and
the magnetic fields.
increasing magnetic field is related to the compression of
the exciton in the ( x , y ) directions and hence to the
increase of f i f , , proportional to RB2.
Lastly, figure 5 illustrates the dependence of the
exciton energy on the size (R,) of the dot. In general, the
energy decreases with increasing dot size. For narrower
dots, only the confinement influences the energy increase,
even in the presence of an external electric or magnetic
field. For intermediate dot radii (R, > 100 A) the
application of an electric field changes the rate of decrease
of ET(Ro)(figure 5(b), (e)) We interpret this feature as
competition between the electric field effect and the
confinement. At large R,, the energy ET(R,) decreases.
The application of a magnetic field does not change
the behaviour of the E,(Ro) curve but increases the
energy value. These same trends can be used to explain
the size-dependent decrease of the ground exciton
oscillator strength in figure 6.
In conclusion, using a variational-perturbative calculation, we have calculated the 1 s exciton properties:
energy and oscillator strength in a parabolic quantum
dot in the presence of parallel electric and magnetic fields.
First, we have tested the validity of the approximation
and compared our results with calculations obtained by
the conventional variational method. The application of
an electric field induces a spatial separation ofthe carriers.
leading to a decrease in the energy and the oscillator
strength. The application of a magnetic field leads to
additional confinement, which induces an enhancement
1779
S Jaziri and A Bennaceur
in the exciton energy a n d oscillator strength. In the case
of narrower dots, we have found that the exciton
properties are little affected by the electric a n d magnetic
fields. The influence of the external fields is greater on
exciton properties for wider quantum dots ( R , > 100
A).
Acknowledgments
The authors would like t o thank Gkrald Bastard a n d
Robson Ferreira for their fruitful discussions and help.
References
[l] Petroff P M, Cibert J, Gossard A C, Dolan G J and Tu
C W 1987 J . Vuc. Sci. Technol. B 5 1204
Cibert J, Petroff P M, Dolan G J, Pearton S J,
Gossard A C and English J H 1986 Auol. Phvs.
Lett. 49 1275
Temkin H. Dolan G J. Panish M B and Chu S N G
1987 Appl. Phys. Lett. 50 413
Kash K, Scherer A, Worlock J M, Craighead H G
and Tdmargo M C 1986 Appl. Phys. Lett 49 1043
Arnot H E G, Watt M, Sotomayor Torres C M,
Glew R, Cusco R, Bates J and Beaumont S P 1989
Superlatt. Microsrruet. 5 459
[21 Hu Y Z, Lindberg M and Koch S W 1990 Phys. Reo. B
42 1713
Bryant G. W 1988 Phys. Rev. B 37 8763
Bryant G W 1992 Excitons in Coilfined Systems ed A
DAndrea et al (New York American Institute of
Physics)
Takagahara 1992 Suf$ Sci. 267 310
Kdyanuma Y 1991 Phys. Reu. B 44 13085
I
..
~~
1780
E31 Kash K, Scherer A, Worlock J M, Craighead H G and
Tamargo M C 1986 Appl. Phys. Lett. 49 1043
Reed M A, Bate R T, Bradshaw K, Duncan W M,
Frensley W R, Lee J W and Shih H D 1986
J. Vac. Sci. Technol. B 4 358
[41 Kohn W 1961 Phys. Rev. 123 1242
Peters F M 1990 P h p ; Re!,, B 42 1486
Maksym P A and Chdkraborty T 1990 Phys. Rev.
Lett. 65 108
Demsey J, Johnson N F, Brey L and Halperin H I
1990 Phys. Rev. B 42 11708
Brey L, Johnson N F and Halperin B I 1989 Phys.
Rev. B 40 10647
[SI SikorskiChandMerktU 1989Phys.Rev.Lett.622164
[6] Weiming Que 199%Solid State Commun. 81 721
Johnson N F and Payne M C 1991 Phys. Rev.
L e t t 67 1157
[7] Halonen V, Chakraborty T and Pietilainen P 1992
Phys. Reu. 45 5986
[XI Chiba Y and Ohnishi S 1988 Phys. Rev. B 38 12988
191 Jaziri S, Bastard G and Bennaceur R 1993 Semicond.
Sci. Technol. 8 670
Cl01 Bockelmann, U, Brunner K and Abstreiter G 1993 Proc.
6th Int. Con$ on Modulated Semiconductor Struetires
[lll Brunner K, Bockelmann U, Abstreiter G, Walther M,
Bohm G. Trankle G and Weimann G 1992 Phys.
Reo. Lett. 69 3216
1121 Jaziri S, Bastard G and Bennaceur R 1993 3rd CunJ on
Optics of Excitons in Confined Systems: 1993
J . Physique IV 3 367
U31 Lee Y C, Mei W N and Liu K C 1982 J. Phys. C: Solid
State Phys. 15 L469
Dai C M, Pei I H and Chuu D S 1990 Physica B
160 317
Huang W J, Chou W C, Chuu D S, Hdn C S and
Mei W N 1988 Semicond. Sci. Technol. 3 202
[14] Jaziri S 1994 Solid State Commuii. 91 171